1. Field of the Invention
The present invention relates to an apparatus for detecting displacement of optical image on an array of photosensitive elements.
2. Description of the Prior Art
The applicant of the present application has already proposed an apparatus for producing an electric signal the phase of which varies depending upon the direction as well as the amount of displacement of an optical image projected on an array of photosensitive elements when the optical image is displaced in the direction in which photosensitive elements are arranged. The apparatus is the subject of U.S. patent application Ser. No. 959,918, now U.S. Pat. No. 4,218,623, the counterpart of which is German patent application No. 2,848,874.b-51. The apparatus mentioned above will be described hereinafter with reference to FIG. 1.
In FIG. 1, the reference numeral 1 designates a photosensitive element array composed of a number (in the shown case the number is eight) of photosensitive elements P.sub.1 -P.sub.8. The photosensitive element array 1 is disposed on or close to a focal plane of an image forming optical system not shown and an optical image of an object is to be focused on the array. From the photosensitive element array 1 are issued photo-electric outputs of the respective elements P.sub.1 -P.sub.8 or the related electric outputs v.sub.1 -v.sub.8. Designated by 3 is a vectorizing circuit comprising eight multipliers 3a-3h. These multipliers 3a-3h multiply the electric outputs v.sub.1 -v.sub.8 by vectors ##EQU1## respectively. As shown in FIG. 2, these vectors are phase-shifted by (2.pi./8) one to another sequentially with the total completing one period.
An adder circuit 5 adds the outputs from the respective multipliers 3a-3h together to form the following added output V: ##EQU2##
This added output V, as described in detail in the prior applications mentioned above, includes information of a spatial frequency component whose spatial period is d, that is, the length of the array of elements P.sub.1 -P.sub.8, and information of other spatial frequency components in the vicinity of the above spatial frequency component.
The above relation may be generalized as follows:
Photosensitive elements, the number of which is M in total, are set in array to produce the respective electric outputs. If one multiplies the electric outputs by vectors which are phase-shifted by (2.pi./N) one to another sequentially and then adds the products together, then one can obtain information of a spatial frequency component having a spatial period corresponding to the length of N photosensitive elements in the direction of array, and information of other spatial frequency components existing near the spatial frequency component.
When an optical image on the array 1 is displaced in any direction, the phase of the added output V will change in the following manner:
Let V' denote the added output obtained when the optical image on the array 1 is displaced by a distance corresponding to one photosensitive element leftward, then V' is given by ##EQU3##
A transformation of the above equation gives: ##EQU4## From the relation, ##EQU5##
In the above equations, v.sub.1 is the electric output related to the intensity of light coming from the area of the array 1 due to the displacement of the optical image and v.sub.2 is the electric output related to the intensity of light newly introduced into the area of the array 1 as the result of the displacement.
As seen from the above, if the second term ##EQU6## in the equation (2) is negligible relative to the first term, then the added output V' after the displacement of optical image, differs from the added output V before the displacement, only in that the phase of the former gets delayed from the latter by (2.pi./8) as will be understood from the equations (1) and (2).
In other words when the above second term is negligible, a leftward or rightward displacement of the optical image by a distance corresponding to one photosensitive element brings about a change in phase of the added output by a constant value -(2.pi./8) or +(2.pi./8).
However, here, note should be taken of the ratio of M to N. As described above, M is the total number of photosensitive elements constituting a photosensitive element array and in the shown case M is 8. N is the number of photosensitive elements corresponding to the spatial length d and in the shown case N is 8. When the ratio is small, for example, 1, 2 or 3, it is impossible to neglect the above-mentioned second term and the change of phase relative to the displacement of optical image can not be constant. To make it constant, M must be about 10 times larger than N. But, the use of such a large ratio of M:N brings forth another problem. Namely, in such case, information containable in the added output V is limited to only those of the spatial frequency component of the spatial period d and of the spatial frequency component very near the above component. Therefore, for an optical image containing little of such spatial frequency component of the spatial period d it becomes impossible to detect the displacement of the optical image. This means that optical images for which the apparatus can detect the displacement are undesirably limited to a great extent.